EARTH'S INTERIOR ADAMS AND WILLIAMSON 243 



bility of the earth at the surface and in the interior, and there would 

 be no need to postulate a heavy material at the center, the earth, on 

 this basis, consisting throughout of silicate rock like that found at 

 the surface. There is no a priori reason why this could not be so, 

 but clearly other lines of evidence must be examined before an 

 answer to this question can be secured. 



MOMENT OF INERTIA OF THE EARTH 



It is obvious that for a given mass (or for a given mean density) 

 the moment of inertia depends on the distribution of density s ; e. g., 

 if there is heavy material at the center and light material at the 

 surface the moment of inertia would be considerably less than if the 

 central density were smaller than that of the surface. 



It is interesting in this connection to recall the old puzzle " How to 

 distinguish between two hollow shells, one of gold, the other of silver, 

 if their diameters and masses be alike, and both painted." 6 Since 

 gold is denser than silver, the volume of the gold shell is less than 

 that of the silver shell, and therefore, on the whole, its mass is 

 farther from the center and its moment of inertia greater. Hence 

 to decide which is the gold and which the silver sphere, it suffices 

 to compare their moments of inertia. This may be done by allowing 

 them to roll down a rough plane, whereupon the gold sphere will 

 move at the slower speed. In an analogous manner, the moment of 

 inertia of the earth may be used to decide which of two proposed 

 distributions of matter within the earth is the more plausible. The 

 moment of inertia itself is not sufficient to fix the density distribu- 

 tion; it can be used, however, as an important check on a density 

 curve deduced from other considerations. The moment of inertia of 

 the earth about the polar axis is known to be close to 8.06 X10 44 

 g.-cm 2 . Since the moment about the equatorial axis differs from 

 that about the polar axis by only one-third of 1 per cent, very little 

 error is introduced by dealing with a sphere of radius equal to the 

 mean radius of the earth and having a moment of inertia equal to the 

 value just mentioned. 



The moment of inertia of the earth if of uniform density from 

 surface to center would be 9.7X10 44 , significantly higher than the true 

 value. In other terms, the moment of inertia of the earth is that 

 of a homogeneous sphere of density 4.6. From this fact, also, fol- 

 lows the qualitative conclusion that in general the density must in- 



« The moment of inertia of a sphere with its mass symetrically distributed about the center is 



In which p is the density at distance r from the center. For a homogeneous sphere this becomes 



§*■ 

 "16 



C-% P r»- 0.4 M r» 



M being the total mass. 

 •See P. Q. Tait, Dynamics, London, 1895. 



